Just curious. Would it be possible to prove the homeomorphism partwithout observing that a set of all natural number sequences with anidentical initial segment is open in N^N, that all such sets form abase for its topology, that continued fraction theory shows that thesesets are mapped to open intervals of irrationals and, finally, thatthere are sufficiently many of these open intervals to provide a basefor the neighborhood systems of all irrationals involved?